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Asymptotically Euclidean Solutions of the Constraint Equations with Prescribed Asymptotics

Lydia Bieri, David Garfinkle, James Isenberg, David Maxwell, James Wheeler

TL;DR

This work develops a rigorous framework to construct asymptotically Euclidean vacuum initial data for General Relativity via the conformal method with prescribed asymptotics. By analyzing the momentum constraint with the vector Laplacian $P_{g,N}$ and the Hamiltonian constraint via the Lichnerowicz equation, it identifies how seed data $(g,A,N)$ control the leading and subleading decay rates, the ADM momentum, and the possible inclusion of anisotropic mass terms. The authors establish Fredholm properties, define momentum- and mass-carrier structures, and prove existence/uniqueness across decay regimes, including threshold behavior, enabling type (CK), (B), and (A) data. A numerical Brill wave example confirms the feasibility of type (A) data and shows that Strominger’s antipodal symmetry conjecture does not hold in this broader asymptotic setting, emphasizing the sensitivity of asymptotic structure to the chosen falloff class. Overall, the results provide a practical blueprint for constructing and analyzing initial data with tailored asymptotics and have implications for the study of null infinity and related conjectures.

Abstract

We demonstrate that in constructing asymptotically flat vacuum initial data sets in General Relativity via the conformal method, certain asymptotic structures may be prescribed a priori through the specified seed data, including the ADM momentum components, the leading- and next-to-leading-order decay rates, and the anisotropy in the metric's mass term, yielding a recipe to construct initial data sets with desired asymptotics. We numerically construct a simple explicit example of an initial data set, with stronger asymptotics than have been obtained in previous work, such that the evolution of this initial data set does not exhibit the conjectured antipodal symmetry between future and past null infinity.

Asymptotically Euclidean Solutions of the Constraint Equations with Prescribed Asymptotics

TL;DR

This work develops a rigorous framework to construct asymptotically Euclidean vacuum initial data for General Relativity via the conformal method with prescribed asymptotics. By analyzing the momentum constraint with the vector Laplacian and the Hamiltonian constraint via the Lichnerowicz equation, it identifies how seed data control the leading and subleading decay rates, the ADM momentum, and the possible inclusion of anisotropic mass terms. The authors establish Fredholm properties, define momentum- and mass-carrier structures, and prove existence/uniqueness across decay regimes, including threshold behavior, enabling type (CK), (B), and (A) data. A numerical Brill wave example confirms the feasibility of type (A) data and shows that Strominger’s antipodal symmetry conjecture does not hold in this broader asymptotic setting, emphasizing the sensitivity of asymptotic structure to the chosen falloff class. Overall, the results provide a practical blueprint for constructing and analyzing initial data with tailored asymptotics and have implications for the study of null infinity and related conjectures.

Abstract

We demonstrate that in constructing asymptotically flat vacuum initial data sets in General Relativity via the conformal method, certain asymptotic structures may be prescribed a priori through the specified seed data, including the ADM momentum components, the leading- and next-to-leading-order decay rates, and the anisotropy in the metric's mass term, yielding a recipe to construct initial data sets with desired asymptotics. We numerically construct a simple explicit example of an initial data set, with stronger asymptotics than have been obtained in previous work, such that the evolution of this initial data set does not exhibit the conjectured antipodal symmetry between future and past null infinity.
Paper Structure (8 sections, 33 theorems, 168 equations, 2 figures)

This paper contains 8 sections, 33 theorems, 168 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and The Conformal Method
  3. The Momentum Constraint
  4. The Hamiltonian Constraint
  5. Constructions
  6. A Numerical Example
  7. Conclusion
  8. Fredholm Properties of the Vector Laplacian

Key Result

Lemma 2.4

Suppose $1<p<\infty$, $k_1,k_2,j\in\mathbb Z_{\geq 0}$ and $\delta_1, \delta_2 \in\mathbb R$. Pointwise multiplication of functions in $C^\infty_c(\mathbb R^n)$ extends to a continuous bilinear map $W^{k_1,p}_{\delta_1}\times W^{k_2,p}_{\delta_2} \rightarrow W^{j,p}_{\delta_1 + \delta_2}$ if In particular, we may choose the optimal value $j = \min(k_1,k_2)$ so long as $\max(k_1, k_2) > n/p$.

Figures (2)

  • Figure 1: $\varphi$ with $q$ given by equation (\ref{['qformula2']}) with ${a_0}=1, \, {r_0}=10, \, \gamma = 2$
  • Figure 2: ${r^3}\rho$ as a function of $\theta$ for $r=10000$ and $r=15000$ with $q$ given by equation (\ref{['qformula2']}) with ${a_0}=1, \, {r_0}=10, \, \gamma = 2$

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • ...and 49 more